‫‪‬‬

‫‪‬‬
‫‪ -3‬دوران ﻣﻨﻄﻘﺔ ﻧﺎﺗﺠﺔ ﻋﻦ ﺗﻘﺎﻃﻊ ﻣﻨﺤﻨﻰ )‪ f(x‬ﻣﻊ ﻣﺴﺘﻘﯿﻢ ‪y=c‬‬ ‫‪ - 2‬دوران ﻣﻨﻄﻘﺔ ﻧﺎﺗﺠﺔ ﻋﻦ ﺗﻘﺎﻃﻊ ﻣﻨﺤﻨﻰ )‪ f(x‬ﻣﻊ ﻣﺴﺘﻘﯿﻢ ‪ y = c‬ﺣﻮل‬
‫ﺣﻮل ﻣﺤﻮر ‪y = c‬‬ ‫ﻣﺤﻮر ‪x‬‬ ‫‪ – 1‬ﻣﻨﺤﻨﻰ ﺣﻮل ﻣﺤﻮر ‪x‬‬
‫‪y‬‬ ‫‪y‬‬
‫‪y‬‬
‫‪3‬‬

‫‪3‬‬
‫‪2‬‬
‫‪y=c‬‬
‫‪f(x) = x2‬‬
‫)‪y = f(x‬‬
‫‪1‬‬

‫‪2‬‬ ‫‪y=c‬‬ ‫‪x‬‬

‫‪x‬‬

‫‪-3‬‬ ‫‪-2‬‬ ‫‪-1‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬

‫‪-1‬‬

‫‪1‬‬
‫)‪y = f(x‬‬ ‫‪-2‬‬

‫ﻧﺼﻒ ﻗﻄﺮ اﻟﻤﻘﻄﻊ ‪r = x2 :‬‬

‫‪-3‬‬
‫‪x‬‬
‫‪-1‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬

‫ﻧﻘﺎﻁ ﺍﻟﺘﻘﺎﻃﻊ ﺗﺄﰐ ﻣﻦ ‪f(x) = c‬‬

‫ﻣﺴﺎﺣﮥ اﻟﻤﻘﻄﻊ ‪A(x) = π(x 2) 2 :‬‬
‫‪-1‬‬

‫=‬ ‫( ∫‬ ‫) ) ( ‪−‬‬ ‫ﺍﳊﺠﻢ ‪:‬‬

‫=‬ ‫)) ( ‪∫ ( −‬‬ ‫ﺍﳊﺠﻢ ‪:‬‬ ‫=‬ ‫)) ( ( ∫‬ ‫ﺣﺠﻢ اﻟﺠﺴﯿﻢ ‪:‬‬

‫‪y‬‬
‫اﻟﻤﻘﻄﻊ اﻟﻨﺎﺗﺞ ﻋﻦ اﻟﺪوران ﻋﺒﺎرة ﻋﻦ داﺋﺮة ﻧﺼﻒ ﻗﻄﺮﻫﺎ = )‪f(x‬‬
‫‪2‬‬
‫‪y‬‬
‫‪y=c‬‬
‫)‪f(x‬‬
‫دوران‬
‫)‪f(x‬‬
‫‪1‬‬

‫‪y‬‬ ‫‪y‬‬

‫‪x‬‬ ‫ﺣﻮل‬ ‫)‪g(x‬‬

‫‪4‬‬ ‫‪3‬‬

‫‪-1‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪2‬‬

‫ﻣﺤﻮر‬ ‫‪x‬‬
‫‪2‬‬ ‫‪1‬‬ ‫‪y=x‬‬
‫‪x‬‬
‫‪-1‬‬ ‫‪1‬‬
‫‪-3‬‬ ‫‪-2‬‬ ‫‪-1‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬

‫‪x‬‬

‫أﻓﻘﻲ‬
‫‪-3‬‬ ‫‪-2‬‬ ‫‪-1‬‬ ‫‪1‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪4‬‬ ‫‪-1‬‬

‫‪y = -2‬‬ ‫‪-1‬‬ ‫‪-2‬‬

‫‪-2‬‬

‫ﻧﻘﺎﻁ ﺍﻟﺘﻘﺎﻃﻊ ﺗﺄﰐ ﻣﻦ )‪f(x) = g(x‬‬

‫‪-2‬‬
‫‪-3‬‬

‫ﺍﳊﺠﻢ ∶‬
‫‪r = √9 −‬‬ ‫ﻧﺼﻒ ﻗﻄﺮ اﻟﻤﻘﻄﻊ ‪:‬‬ ‫ﻧﺼﻒ ﻗﻄﺮ اﻟﻤﻘﻄﻊ ‪r = x :‬‬
‫=‬ ‫)‪∫ (( + 2) − ( ( ) + 2‬‬
‫=‬ ‫) ) ( ‪∫ ( ( ) −‬‬ ‫ﺍﳊﺠﻢ ‪:‬‬
‫=‬ ‫‪∫ (9 −‬‬ ‫)‬ ‫ﺣﺠﻢ اﻟﺠﺴﯿﻢ ‪:‬‬ ‫=‬ ‫∫‬ ‫ﺣﺠﻢ اﻟﺠﺴﯿﻢ ‪:‬‬

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